This has not been my experience. I prefer to climb at a sustainable speed so I never face lactic acid buildup or muscle tightening. I am in normal, not exceptional shape. If I climbed at lactic threshold like you are describing, I'd be dead after a few days.
I’m not talking about doing anaerobic efforts. Just normal climbing. But that does require more effort than pedaling downhill even at high speeds. Putting in that kind of effort leads to making lactic acid as does any exercise. But stopping pedaling at the top of a hill and coasting for extended periods after a hard effort doesn’t clear the lactic acid from the muscles and allows them to stiffen. The stiffness is even worse when the muscles are locked into one position which is what most people do when coasting.